In mathematics — specifically, in functional analysis — an Asplund space or strong differentiability space is a type of well-behaved Banach space. Asplund spaces were introduced in 1968 by the mathematician Edgar Asplund, who was interested in the Fréchet differentiability properties of Lipschitz functions on Banach spaces.
Equivalent definitions
There are many equivalent definitions of what it means for a Banach space X to be an Asplund space:
- X is Asplund if, and only if, every separable subspace Y of X has separable continuous dual space Y∗.
- X is Asplund if, and only if, every continuous convex function on any open convex subset U of X is Fréchet differentiable at the points of a dense Gδ-subset of U.
- X is Asplund if, and only if, its dual space X∗ has the Radon–Nikodým property. This property was established by Namioka & Phelps in 1975 and Stegall in 1978.
- X is Asplund if, and only if, every non-empty bounded subset of its dual space X∗ has weak-∗-slices of arbitrarily small diameter.
- X is Asplund if and only if every non-empty weakly-∗ compact convex subset of the dual space X∗ is the weakly-∗ closed convex hull of its weakly-∗ strongly exposed points. In 1975, Huff & Morris showed that this property is equivalent to the statement that every bounded, closed and convex subset of the dual space X∗ is closed convex hull of its extreme points.